Let G = (V, E) be a directed graph and let P be a shortest path from s to t in G. In the replacement paths problem, we are required to find, for every edge e on P, a shortest path from s to t in G that avoids e. The o...
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Let G = (V, E) be a directed graph and let P be a shortest path from s to t in G. In the replacement paths problem, we are required to find, for every edge e on P, a shortest path from s to t in G that avoids e. The only known algorithm for solving the problem, even for unweighted directed graphs, is the trivial algorithm in which each edge on the path, in its turn, is excluded from the graph and a shortest paths tree is computed from s. The running time is O(mn + n(2) log n). The replacement paths problem is strongly motivated by two different applications: (1) The fastest algorithm to compute the k simple shortest paths between s and t in directed graphs [Yen 1971;Lawler 1972] computes the replacement paths between s and t. Its running time is O(mnk). (2) The replacement paths problem is used to compute the Vickrey pricing of edges in a distributed network. It was raised as an open problem by Nisan and Ronen [2001] whether it is possible to compute the Vickrey pricing faster than n computations of a shortest paths tree. In this article we present the first nontrivial algorithm for computing replacement paths in unweighted directed graphs (and in graphs with small integer weights). Our algorithm is Monte-Carlo and its running time is O(m root n). This result immediately improves the running time of the two applications mentioned above in a factor of root n. We also show how to reduce the problem of computing k simple shortest paths between s and t to O(k) computations of a second simple shortest path from s to t each time in a different subgraph of G. The importance of this result is that computing a second simple shortest path may turn out to be an easier problem than computing the replacement paths, thus, we can focus our efforts to improve the k simple shortest paths algorithm in obtaining a faster algorithm for the second shortest path problem.
A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general grap...
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A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general graph, this problem is #P-complete. We provide a polynomial time algorithm to generate the whole family of mis in a caterpillar graph. We also characterize the independent graph (intersection graph of mis) and the clique graph (intersection graph of cliques) of complete caterpillar graphs. (C) 2011 Elsevier B.V. All rights reserved.
Purpose - To simulate large parasitic resistive networks, one must reduce the size of the circuit models through methods that are accurate and preserve terminal connectivity and network sparsity. The purpose here is t...
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Purpose - To simulate large parasitic resistive networks, one must reduce the size of the circuit models through methods that are accurate and preserve terminal connectivity and network sparsity. The purpose here is to present such a method, which exploits concepts from graph theory in a systematic fashion. Design/methodology/approach - The model order reduction problem is formulated for parasitic resistive networks through graph theory concepts and algorithms are presented based on the notion of vertex cut in order to reduce the size of electronic circuit models. Four variants of the basic method are proposed and their respective merits discussed. Findings - The algorithms proposed enable the production of networks that are significantly smaller than those produced by earlier methods, in particular the method described in the report by Lenaers entitled "Model order reduction for large resistive networks". The reduction in the number of resistors achieved through the algorithms is even more pronounced in the case of large networks. Originality/value - The paper seems to be the first to make a systematic use of vertex cuts in order to reduce a parasitic resistive network.
Let M be the number of edges in a maximum matching in graphs with m edges, maximum vertex degree k and shortest simple odd-length cycle length L. We show that M >= {m/2 - m/2L, if k = 2, m/k - m/(k+L)k, if k > 2...
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Let M be the number of edges in a maximum matching in graphs with m edges, maximum vertex degree k and shortest simple odd-length cycle length L. We show that M >= {m/2 - m/2L, if k = 2, m/k - m/(k+L)k, if k > 2. This lower bound is tight. When no simple odd-length cycle exists it is known previously that M >= m/k.
Hypergraph width measures are important in studying the complexity of constraint satisfaction problems (CSPs). We present a general exact exponential algorithm for a large variety of these measures. As a consequence, ...
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Hypergraph width measures are important in studying the complexity of constraint satisfaction problems (CSPs). We present a general exact exponential algorithm for a large variety of these measures. As a consequence, we obtain algorithms which, for a hypergraph H on n vertices and m hyperedges, compute its generalized hypertree-width in time O*(2(n)) and its fractional hypertree-width in time O(1.734601(n) .m)(3). (C) 2011 Elsevier B.V. All rights reserved.
We consider testing directed graphs Eulerianity in the orientation model introduced in Halevy et al. [2005]. Despite the local nature of the Eulerian property, it turns out to be significantly harder to test than othe...
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We consider testing directed graphs Eulerianity in the orientation model introduced in Halevy et al. [2005]. Despite the local nature of the Eulerian property, it turns out to be significantly harder to test than other properties studied in the orientation model. We show a nonconstant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of 1-sided tests for this property. On the positive side, we give several 1-sided and 2-sided tests, including a sublinear query complexity 2-sided test, for general graphs. For special classes of graphs, including bounded-degree graphs and expander graphs, we provide improved results. In particular, we give a 2-sided test with constant query complexity for dense graphs, as well as for expander graphs with a constant expansion parameter.
We present an algorithm that, given two vertices s(1) and s(2) of a directed acyclic graph, constructs in linear time a data structure using linear space that, for each pair (u, v) of two vertices u and v, in constant...
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We present an algorithm that, given two vertices s(1) and s(2) of a directed acyclic graph, constructs in linear time a data structure using linear space that, for each pair (u, v) of two vertices u and v, in constant time can output a tuple (s(1), t(1), s(2), t(2)) with {t(1), t(2)} = {u, v} such that there are two disjoint paths p(1), from s(1) to t(1), and p(2), from s(2) to t(2), if such a tuple exists. The data structure is simpler than such a data structure for general directed graphs that can be obtained from results of Georgiadis and Tarjan. Disjoint can mean vertex- as well as edge-disjoint. As an application and main result we show that the data structure can be used to solve the 2-disjoint paths problem on directed acyclic graphs optimally, i.e., in linear time. (c) 2012 Elsevier B.V. All rights reserved.
We present I/O-efficient single-source shortest path algorithms for undirected graphs. Our main result is an algorithm with I/O complexity O(root nm log L)/ B + MST(n, m)) on graphs with n vertices, m edges, and arbit...
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We present I/O-efficient single-source shortest path algorithms for undirected graphs. Our main result is an algorithm with I/O complexity O(root nm log L)/ B + MST(n, m)) on graphs with n vertices, m edges, and arbitrary edge lengths between 1 and L;MST(n, m) denotes the I/O complexity of computing a minimum spanning tree;B denotes the disk block size. If the edge lengths are drawn uniformly at random from ( 0, 1], the expected I/O complexity of the algorithm is O(root nm/ B + ( m/ B) log B + MST(n, m)). A simpler algorithm has expected I/O complexity O(root nm log B)/ B + MST(n, m) for uniformly random edge lengths.
In this paper, we define a new subclass of integer linear programming problems called the composition problem. We shall propose efficient algorithms for solving this problem and its variants. Moreover, as an applicati...
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In this paper, we define a new subclass of integer linear programming problems called the composition problem. We shall propose efficient algorithms for solving this problem and its variants. Moreover, as an application of the composition problem, those algorithms are applied to solve the P-constrained secure set problem, which is a variation of the secure set problem introduced in [5], on trees. A P-constrained secure set problem is to find a minimum secure set containing a set of vertical bar P vertical bar predetermined vertices.
We introduce the notions of centrality interference and centrality robustness, as measures of variation of centrality values when the structure of a network is modified by removing or adding individual nodes from/to a...
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We introduce the notions of centrality interference and centrality robustness, as measures of variation of centrality values when the structure of a network is modified by removing or adding individual nodes from/to a network. Centrality analysis allows categorizing nodes according to their topological relevance in a network. Thus,centrality in terf er en c e analysis allows understanding which parts of a network are mostly influenced by a node and, conversely, centrality robustness allows quantifying the functional dependency of a node from other nodes in the network. We examine the theoretical significance of these measures and apply them to classify nodes in a road network to predict the effects on the traffic jam due to variations in the structure of the network. In these case the interference analysis allows to predict which are the distinct regions of the network affected by the function of different nodes. Such notions, when applied to a variety of different contexts, opens new perspectives in network analysis since they allow predicting the effects of local network modifications on single node as well as global network functionality.
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